L’INSTITUTFOURIER DE ANNALES

A L

E S

E D L ’IN IT ST T U

F O U R IE

ANNALES

DE

L’INSTITUT FOURIER

Zbigniew BLOCKI

Weak solutions to the complex Hessian equation Tome 55, n^o5 (2005), p. 1735-1756.

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55, 5 (2005), 1735–1756

WEAK SOLUTIONS

TO THE COMPLEX HESSIAN EQUATION

by Zbigniew BLOCKI

1. Introduction.

For a smooth function u defined on an open subset of Cⁿ and m= 1, . . . , nthe elementary complex Hessian operator is defined by

Hm(u) =

1j1<...<jmn

λj1. . . λjm,

whereλ₁, . . . , λ_n are the eigenvalues of the complex Hessian (∂²u/∂z_j∂z_k).

W e haveH1= ∆/4 andHnis the complex Monge-Amp`ere operator. Using the operatorsd=∂+∂ andd^c=i(∂−∂), so thatdd^c= 2i∂∂, one gets

(dd^cu)^m∧ω^n−m= 4ⁿm!(n−m)!Hm(u)dλ,

where ω =dd^c|z|² is the fundamental K¨ahler form anddλ is the volume form.

The class of smooth admissible functions for the operator H_m is naturally defined by the condition Hm(u+A|z|²) 0 for every A 0.

Using for example approximation by smooth functions one can define this

Keywords: ComplexHessian equation, plurisubharmonic functions.

Math. classification: 32U05, 35J60.

notion also for non-smooth functions. We will denote this class byPmand call such functionsm-subharmonic. We clearly have

P SH=Pn⊂. . .⊂ P1=SH.

The classPmis essentially determined by the following property dd^cu1∧. . .∧dd^cum∧ω^n−m0, u1, . . . , um∈ Pm∩C^∞. It would be interesting to find a geometric characterization of such func- tions. A necessary condition is dd^cu∧ω^m−n 0, which means that u is subharmonic on every complex n−m+ 1-dimensional affine subspace of Cⁿ, but this condition is not sufficient if 1< m < n.

The aim of this paper is to study basic, mostly local properties of m-subharmonic functions and the operatorHm. Similarly as in [4] or [5]

one can introduce the domain of definition of the operatorH_m: a function u∈ Pmis said to belong to the classDmif there is a regular Borel measureµ such that ifu_jis a decreasing sequence of smoothm-subharmonic functions converging tou(we consider only germs of functions,uj may be defined on a smaller domain thanu) thenHm(uj) tends weakly toµ. Foru∈Dm we setH_m(u) =µand one can easily show thatDmis the maximal subclass of Pmwhere the operatorHmcan be extended (the values ofHmare regular Borel measures) so that it is continuous for decreasing sequences. Similarly as in [4] and [5] form=n, we shall completely characterize the class Dm. We will show in particular the following result.

THEOREM 1.1. — If K Ω ⊂ Cⁿ and u, v ∈ Pm(Ω) are such that u∈ Dm(Ω),uv inΩ\K, thenv∈ Dm(Ω).

Theorem 1.1 implies for example that the classDmcontains functions from Pm(Ω) which are locally bounded away from a compact subset of Ω.

One of the main results is the following natural characterization of m-maximal functions (a functionu∈ Pm(Ω), Ω open in Cⁿ, is called m- maximal ifv ∈ Pm(Ω),v uoutside a compact subset of Ω implies that vuin Ω).

THEOREM 1.2. — A function u ∈ Dm is m-maximal if and only if H_m(u) = 0.

Theorem 1.2 implies in particular that m-maximality of locally bounded m-subharmonic functions is a local property. We conjecture that

this is the case without the assumption of boundedness but this is an open problem even ifm=n= 2.

One can check that the function

(1.1) G(z) =

−|z|^2−2n/m, m < n, log|z|, m=n.

is a fundamental solution for the operator Hm (note that G ∈ Dm by Theorem 1.1). We clearly have

(1.2) G∈L^p_loc⇔p < nm

n−m.

This leads naturally to the conjecture that for every p < nm/(n−m) one hasPm⊂L^p_loc. We are only able to show the following partial result.

PROPOSITION 1.3. — For everyp < n/(n−m)we have Pm⊂L^p_loc. Note that we get optimal exponent in the well known, extreme cases m= 1 andm=n.

For m = n we deal with the complex Monge-Amp`ere operator and plurisubharmonic functions and the above results are of course known (see e.g. [1], [2], [5], [11], [18]). The aim of this paper is to concentrate on those problems related to the Hessian operator where the methods of the complex Monge-Amp`ere operator cannot be automatically repeated.

The real Hessian operator has also been studied quite extensively in the recent years – see e.g. [8], [15], [16], [19], [21], [22]. It is clear that if u(z) =u(x+iy) is independent ofy then it is m-subharmonic if and only if it ism-convex (see [22]). This means that in a way the complex Hessian operator is a generalization of the real one and indeed, for example, some results of Section 2 are generalizations of some results from [22].

The real Hessian operator for functions on open subsets of Rⁿ is defined in the same way, one only takes the real Hessian instead of the complex one. Denote the class of m-convex functions by Φm. The fundamental solution for the real Hessian operator is

H(x) =





−|x|^2−n/m, m < n/2, log|x|, m=n/2,

|x|^2−n/m, m > n/2.

One has

H ∈W_loc^1,q⇔q < nm n−m.

It was in fact proved in [22, Theorem 4.1] that Φ_m ⊂ W_loc^1,q for q <

nm/(n−m). From the Sobolev theorem it thus follows that form n/2 and p < nm/(n−2m) we have Φm ⊂L^p_loc, which is an optimal result in terms of the exponentp.

In the complex case however it is not possible to prove optimal L^p estimates for m-subharmonic functions via gradient estimates and the Sobolev theorem. On one hand we have

G∈W_loc^1,q ⇔q < 2nm 2n−m.

But the function u(z) = log|z₁|belongs toPm for everym,u /∈W_loc^1,2 and 2<2nm/(2n−m) if 2mn.

Most of the results of this paper were contained in the first version of the author paper [5]. Later however, a simpler proof of the characterization of the domain of definition of the complex Monge-Amp`ere operator was found, not employing the complex Hessian operator.

The paper is organized as follows. In Section 2 certain basic facts on elementary symmetric functions are collected. In Section 3 we prove the main properties of m-subharmonic functions including a special case of Theorem 1.2 for continuous functions. The proof relies heavily on an a priori estimate for a special case of the complex Hessian equation which is presented in Section 4. We remark that a much more general estimate and a solution of the Dirichlet problem for non-degenerate equation was independently shown in [20] with essentially the same methods as below.

Section 5 is devoted to the characterization of the class Dm. Since most of the proofs are similar to those from [5], they are sketched only briefly.

Finally, in Section 6 we prove Proposition 1.3.

We have concentrated here on the study of weak solutions of complex Hessian equations and therefore we restricted ourselves to the elementary Hessian operatorsH_m- as [22] did for example in the real case. Of course many results could be generalized here to more general complex Hessian operators. The existence of strong solutions in domains in Cⁿ for such equations was recently proved in [20]. In particular, one could study the operator (dd^cu)^m∧ω^n−m on manifolds, where ω is an arbitrary K¨ahler form. This would perhaps be interesting from a geometric point of view.

For the global Dirichlet problem on compact K¨ahler manifolds, in analogy with the case of the Calabi-Yau theorem, one would have to consider the operator (ω+dd^cϕ)^m∧ω^n−m.

Acknowledgements. — The author would like to thank PaweMl Strze- lecki for providing the references [6] and [17] (they are used in the remark at the end of Section 3).

2. Basic properties of elementary symmetric functions.

In this section we recall some basic facts from (multi-)linear algebra needed in the paper. We set

S_m(λ) =

1j1<...<jmn

λ_j1. . . λ_jm, λ= (λ₁, . . . , λ_n)∈Rⁿ.

The elementary symmetric function Smis determined by (λ1+t). . .(λn+t) =

n m=0

Sm(λ)t^n−m, t∈R.

By Γm we denote the closure of the connected component of {Sm > 0} containing (1, . . . ,1). One can show that

Γm={λ∈Rⁿ :Sm(λ1+t , . . . , λn+t)0∀t0}

and, since

Sm(λ1+t , . . . , λn+t) = m p=0

Sp(λ)t^m−p, t∈R, we also have

Γm={S10} ∩. . .∩ {Sm0}.

In particular

Γ_n ⊂. . .⊂Γ₁.

By G˚arding [14] the set Γmis a convex cone inRⁿ andSm^1/mis concave on Γ_m. By Maclaurin inequality on Γ_m one also has

n m

₋1/m

S_m^1/m n

p ₋1/p

S_p^1/p, 1pm.

ByHwe will denote the vector space (overR) of (complex) hermitian n×nmatrices. ForA∈ Hletλ(A) = (λ1, . . . , λn)∈Rⁿbe the eigenvalues ofA. We set

Sm(A) =Sm(λ(A)).

The function Smis determined by det(A+tI) =

n m=0

Sm(A)t^n−m, t∈R.

ThenS_mis a homogeneous polynomial of ordermonHwhich is hyperbolic with respect to I (that is for every A ∈ H the equationSm(A+tI) = 0 hasmreal roots; see [14]). As in [14] we define the cone

Γm:={A∈ H:Sm(A+tI)0∀t0}.

W e have

Γ_m={A∈ H:λ(A)∈Γ_m}={S₁0} ∩. . .{S_m0}.

It was proved in [14] that the cone Γmis convex and the functionSm^1/m is concave onΓ_m.

Let M :H^m→Rbe the polarized form ofS_m - it is determined by the following three properties:M is linear in every variable, symmetric and

M(A, . . . , A) =S_m(A), A∈ H. The inequality due to G˚arding [14, Theorem 5] asserts that

(2.1) M(A1, . . . , Am)Sm(A1)^1/m. . .Sm(Am)^1/m, A1, . . . , Am∈Γm. Real (1,1)-formsβ we associate with hermitian matrices (a_jk) by

β = 2

j,k

a_jkidzj∧dz_k

(so that ω is associated with the identity matrix I). After diagonalizing (a_jk), we see that

β^m∧ω^n−m=m!S_m((a_jk))ωⁿ.

It is also clear thatβ₁∧. . . β_m∧ω^n−mis the polarized form ofβ^m∧ω^n−m. Accordingly, we set

Γ_m:={β ∈C(1,1):β∧ωⁿ⁻¹0, β²∧ωⁿ⁻²0, . . . , β^m∧ω^n−m0}.

The crucial fact for us will be the following property.

PROPOSITION 2.1. — Forβ1, . . . , βp∈Γm,pm, we have β₁∧. . .∧β_p∧ω^n−m0.

Proof. — We need to show that for any (1,0) formsα₁, . . . , α_m−p we have

iα1∧α1∧. . .∧iαm−p∧αm−p∧β1∧. . .∧βq∧ω^n−m0.

But since iαj ∧αj ∈ Γn ⊂ Γm (this is because (iαj ∧αj)² = 0), we may assume that p= m. Then the proposition follows from the G˚arding

inequality (2.1).

ForB∈ Hwe define Dm(B) :=

∂Sm

∂b_pq(B)

∈ H.

We then have

tr(AD_m(B)) =mM(A, B, . . . , B), in particular

(2.2) tr(BD_m(B)) =S_m(B).

IfB is diagonal then so is Dm(B). Ifλ=λ(B) then λ(D_m(B)) = (S_m−1(λ⁽¹⁾), . . . , S_m−1(λ⁽ⁿ⁾)),

where λ^(j)= (λ1, . . . , λj−1, λj+1, . . . , λn). IfB ∈Γmthen fort >0 S_m−1(λ^(j)) =t⁻¹

S_m(λ+ (0, . . . , t , . . . ,0))−S_m(λ)

0

so thatD_m(B)0. By (2.1) we have

(2.3) tr(AD_m(B))mS_m(A)^1/mS_m(B)^(m−1)/m, A, B∈Γ_m, and

Sm(A)^1/m= 1

m inf{M(A, B, . . . , B) :B∈Γm, Sm(B)1}

= 1

m inf{tr(ADm(B)) :B∈Γm, Sm(B)1}, A∈Γm.

3. The m-subharmonic functions and the complex Hessian operator.

In this section we define the class of admissible functions for the complex Hessian operator H_m and prove their basic properties. Most of the proofs are the same as in the case of plurisubharmonic functions and the Monge-Amp`ere operator (that is when m =n) and therefore we will present them only briefly.

A function u is called m-subharmonic (we write u ∈ Pm) if it is subharmonic and

dd^cu∧β₁∧. . .∧β_m−1∧ω^n−m0, β₁, . . . , β_m−1∈Γ_m.

The following basic properties of m-subharmonic functions either follow immediately from Proposition 2.1 or can be proven in the same way as in the classical case, and therefore their proofs are left to the reader.

PROPOSITION 3.1. — i) Ifu isC² smooth then it ism-subharmonic if and only if the form dd^cubelongs pointwise toΓm;

ii) Ifu, v∈ Pmthenu+v∈ Pm;

iii) If u ∈ Pm and γ : R →R is a convex, increasing function then γ◦u∈ Pm;

iv) Ifuism-subharmonic then the standard regularizationsu∗ρεare alsom-subharmonic ;

v) If {uι} ⊂ Pm is locally uniformly bounded from above then (sup_ιu_ι)^∗∈ Pm, wherev^∗ denotes the upper regularization ofv;

vi)P SH =Pn⊂. . .⊂ P1=SH.

The next result was proven in [23] form=n.

PROPOSITION 3.2. — For a bounded domainΩ inCⁿ and f ∈C(Ω) set

u:= sup{v∈ Pm(Ω) :vf}.

Assume moreover that u^∗=u_∗=f on∂Ω. Thenu∈ Pm(Ω)∩C(Ω).

Proof. — It is clear that u^∗ ∈ Pm(Ω) and u^∗ f, thus u = u^∗ is upper semi-continuous in Ω. To show the lower semi-continuity in Ω fix z0∈Ω andε >0. From the uniform continuity of f on Ω and ofu=f on

∂Ω it follows that we can find δ >0 such that

(3.1) dist (z₀, ∂Ω)2δ,

(3.2) z∈Ω, w∈∂Ω, |z−w|3δ ⇒ |u(z)−f(w)|ε,

(3.3) z, z ∈Ω, |z−z|δ ⇒ |f(z)−f(z)|2ε.

Fixz with|z0−z|δ. Forz∈Ω set v(z) :=

max{u(z+z0−z)−2ε, u(z)}, dist (z, ∂Ω)δ,

u(z), dist (z, ∂Ω)< δ.

If dist (z, ∂Ω)2δthen we can findw∈∂Ω with|w−z|2δ. Using (3.2) twice we get

u(z+z0−z)f(w) +εu(z) + 2ε.

This implies thatv(z) =u(z) if dist (z, ∂Ω)2δ, and thusv∈ Pm(Ω). On the other hand, if dist (z, ∂Ω)δ, then by (3.3)

u(z+z0−z)f(z+z0−z)f(z) + 2ε and it follows that vf in Ω. Thereforevuin Ω and by (3.1)

u( z)v( z)u(z0)−2ε,

henceuis lower semi-continuous.

Proposition 3.2 will mostly be used in the situation when Ω is a regular domain (with respect to harmonic functions) andf is harmonic in

Ω. In such a case, from the maximum principle it follows that the condition vf in the definition ofuis equivalent tov^∗f on∂Ω.

For continuousm-subharmonic functions we can inductively define a closed nonnegative current

(3.4) dd^cu1∧. . . dd^cup∧ω^n−m:=dd^c

u1dd^cu2∧. . .∧dd^cup∧ω^n−m , u1, . . . , up∈ Pm∩C, pm.

(We have used the fact that the coefficients of nonnegative currents are complex measures, see e.g. [13].) We can also define a nonnegative current (3.5)

d(u₀−u₁)∧d^c(u₀−u₁)∧dd^cu₂∧. . .∧dd^cu_p∧ω^n−m

u₀, u₁, . . . , u_p∈ Pm∩C, pm.

as follows. We note that

d(u₀−u₁)∧d^c(u₀−u₁) = 2du₀∧d^cu₀+2du₁∧d^cu₁−d(u₀+u₁)∧d^c(u₀+u₁) and

du∧d^cu= 1

2dd^c(u+C)²−(u+C)dd^cu, u∈ Pm∩C, where Cis sufficiently big, and use the previous part.

The proofs of the following three results form=ncan be essentially found in [1].

PROPOSITION3.3. — The operators (3.4) and (3.5) are continuous for locally uniformly convergent sequences inPm∩C.

Proof. — It is enough to prove the continuity of the operator (Pm∩C)^p(u1, . . . , up)−→u1dd^cu2∧. . .∧dd^cup∧ω^n−m. This follows inductively from the fact that the coefficients of a nonnegative current are complex measures and since the convergence is uniform.

PROPOSITION 3.4. — Foru, v∈ Pm∩Cwe have

(dd^cmax{u, v})^m∧ω^n−mχ_{u>v}(dd^cu)^m∧ω^n−m+χ_{uv}(dd^cv)^m∧ω^n−m, where χ_A denotes the characteristic function of a setA.

Proof. — For a compactK⊂ {u=v}by Proposition 3.3 we have

K

(dd^cmax{u, v})^m∧ω^n−mlim

ε↓0

K

(dd^cmax{u, v+ε})^m∧ω^n−m

=

K

(dd^cv)^m∧ω^n−m.

PROPOSITION 3.5. — Let Ω be a bounded domain in Cⁿ, u, v ∈ Pm(Ω) ∩ C(Ω) are such that u v on ∂Ω and (dd^cu)^m ∧ ω^n−m (dd^cv)^m∧ω^n−min Ω. Thenuv inΩ.

Proof. — Suppose that {u > v} =∅. We can then find ε > 0 such that S:={ u > v} =∅, whereu:=u+εψandψ(z) =|z|²−M is negative in Ω. We have

(dd^c(u+εψ))^m∧ω^n−m(dd^cu)^m∧ω^n−m+ε^mωⁿ

and from Proposition 3.4 it follows that (dd^cu)^m∧ω^n−m(dd^cv)^m∧ω^n−m in Ω. However, sinceu=vnear∂Ω, regularizingu, v and using the Stokes theorem we get

Ω

(dd^cu)^m∧ω^n−m=

Ω

(dd^cv)^m∧ω^n−m

and we must thus have (dd^cu)^m∧ω^n−m= (dd^cv)^m∧ω^n−min Ω. On the other hand

S

(dd^cu)^m∧ω^n−m

S

(dd^cu)^m∧ω^n−m+ε^m

S

ωⁿ

and we get a contradiction.

A function u ∈ Pm(Ω), Ω open in Cⁿ, is called m-maximal if v ∈ Pm(Ω), v u outside a compact subset of Ω implies that v u in Ω. We first prove Theorem 1.2 for continuous functions.

THEOREM 3.6. — A function u ∈ Pm∩C(Ω) is m-maximal if and only if it solvesHm(u) = 0.

Theorem 3.6 will easily follow from the comparison principle (Propo- sition 3.5) and the solution of the Dirichlet problem for the m-Hessian equation in a ball.

THEOREM3.7. — LetB be a ball inCⁿ andϕa continuous function on∂B. Then the following Dirichlet problem





u∈ Pm(B)∩C(B)

(dd^cu)^m∧ω^n−m= 0 in B u=ϕ on ∂B

has a unique solution.

Proof. — Uniqueness is a consequence of Proposition 3.5. To show the existence we first assume that ϕ is smooth and for a constant a > 0 consider the Dirichlet problem

(3.6)





u∈ Pm(B)∩C^∞(B)

(dd^cu)^m∧ω^n−m=aωⁿ in B u=ϕ on ∂B.

By the Evans-Krylov theory (see e.g. [7, Theorem 1]) there exists a solution of (3.6) provided that we have an a priori bound

(3.7) ||u||_C1,1(B)C,

whereC depends only onaandϕ. The proof of this estimate is postponed to Section 4.

Assuming that (3.7) is proven, and thus that we can solve (3.6), letϕ be arbitrary continuous. Approximate it from below byϕ_j∈C^∞(∂B). Let uj be a solution of (3.6) withϕj anda= 1/j. Let ψ(z) =|z−z0|²−R², where z₀ is the center and R the radius of B. For k j Proposition 3.5 gives

uk+j^−1/mψ− ||ϕj−ϕ||L^∞(∂B)uj uk.

This implies that u_j converges uniformly on B to a certainu, which is a

solution by Proposition 3.3.

Proof of Theorem 3.6. — Proposition 3.5 directly implies that if u satisfies Hm(u) = 0 then it is maximal. On the other hand, assume that uis maximal and let B Ω be a ball. By Theorem 3.7 we findu∈C(Ω) determined byu=uin Ω\B,u∈ Pm(B) and (dd^cu)^m∧ω^n−m= 0 inB.

By the comparison principle again we haveuuinBand thusu∈ Pm(Ω).

Sinceuis maximal, it follows thatu=uand we getH_m(u) = 0.

Remark. — For m = n Theorem 3.7 was proved by Bedford and Taylor [1] with the help of an interior C^1,1 estimate ([1, Theorem 6.7]), which, together with later simplifications due to Demailly [11], gives an overall simpler and more elementary proof than the one presented here (not employing strong solutions at all and thus not using the Evans- Krylov theory and estimate (3.7)). It relied however on the following, rather rare, property: the group of smooth diffeomorphisms of the unit ball in Cⁿ preserving plurisubharmonic functions is transitive. Note that this not true in the real case (where plurisubharmonic functions are replaced by the convex ones - then we only have the affine mappings) and one can also show that it is not true for m = 1, that is for subharmonic functions. One can namely check that in this case such a diffemorphism F = (F¹, . . . , F²ⁿ) has to satisfy two properties: 1) the (real) Jacobian matrix of F is orthogonal at every point; 2) ∆F^j = 0,j = 1, . . . ,2n. By the Liouville theorem (see e.g. [6] or [17]) the mappings satisfying 1) are precisely the M¨obius transformations. However, the Kelvin transformation z → z/|z|² is harmonic only in the real dimension 2. Thus, if n >1 the mappings satisfying 1) and 2) are precisely linear M¨obius transformations and the group in question is not transitive. We suspect that it is also not transitive if 1< m < n.

4. The a priori estimate.

In this section we will prove the estimate (3.7). We essentially follow [8] using some ideas from [7] and the simplification from [21]. We use the notation uj =∂u/∂zj, u_j =∂u/∂zj. The real partial derivatives ofuwill be denoted byuxj,uyj, and byuζ we mean the derivative ofuin directionζ.

It is no loss of generality to consider the equation

(4.1) S_m((u_jk)) = 1.

Computing the derivative of both sides of (4.1) in a direction ζwe get

(4.2) a^jku_ζjk= 0,

where

a^pq

=Dm((u_jk)) =

∂Sm

∂upq

((u_jk))

.

By Section 2 we have (a^jk)>0. If (u_jk) is diagonal then so is (a^jk) which implies that the product of these is a hermitian matrix. This means that for every p, q

(4.3) a^pku_qk =a^jqu_jp, and by (4.2)

(4.4) a^jk[zpuq−zqup]_jk= 0.

Since Sm^1/m is concave on Γm it follows that so is G := logSm. Differentiating the logarithm of both sides of (4.1) twice in direction ζ we get

j,k,p,q

∂²G

∂u_jk∂upq

u_jkζu_pqζ+

j,k

∂G

∂u_jku_jkζζ= 0.

The concavity of Gimplies that the first term is nonpositive and we get

(4.5) a^jku_ζζjk 0.

It is no loss of generality to assume that B = B(0,1) is the unit ball in Cⁿ and that ϕ∈ C^∞(B) is harmonic in B. ByC we will denote possibly different constants depending only on||ϕ||_C3,1(B)and say that they are under control. We also setψ(z) := (|z|²−1)/2. From the comparison principle we get, for sufficiently bigC,ϕ+Cψuϕ. This coupled with (4.2) gives

(4.6) ||u||_C0,1(B)C.

We now turn to the estimates of D²uon∂B. Forζ ∈∂B bys, t we will denote the (real) tangential directions atζand byN the outer normal direction. We clearly have

(4.7) u_st=ϕ_st+ (u−ϕ)_Nδ_st. From (4.6) it follows therefore that

(4.8) |u_st(ζ)|C, ζ∈∂B.

Next we estimate the mixed tangential-normal derivativeu_tN(ζ⁰) for a fixed ζ⁰∈∂B. We may assume that ζ0 = (0, . . . ,0,1), so that at ζ⁰ we have N =∂/∂xn. First assume thatt=∂/∂xpfor some pn−1. Set

v: = 2 Re [z_p(u−ϕ)_n−z_n(u−ϕ)_p]

=x_p(u−ϕ)_xn−x_n(u−ϕ)_xp+y_p(u−ϕ)_yn−y_n(u−ϕ)_yp. Thenv= 0 on∂B,|v|C on∂B(ζ⁰,1)∩B and by (4.4)

±a^jkv_jk−C

j

a^jj.

We now consider the barrier function w :=±v−C1|z−ζ0|²+C2ψ. W e can choose constants 0 C₁ C₂ under control so that w 0 on

∂(B ∩B(ζ⁰,1)) and a^jkw_jk 0 in B ∩B(ζ⁰,1). Therefore w 0 in B∩B(ζ⁰,1),

|v|C1|z−ζ0|²−C2ψ

and it follows that |v_xn(ζ⁰)|C. Atζ⁰ we have however v_xn=−(u−ϕ)_xp−(u−ϕ)_xpxn and thus|uxpxn(ζ⁰)|C.

To estimateu_ypxn(ζ⁰) we take v: = 2 Im [zp(u−ϕ)n−zn(u−ϕ)p]

=y_p(u−ϕ)_xn−x_n(u−ϕ)_yp+y_n(u−ϕ)_xp−x_p(u−ϕ)_yn. and proceed similarly. Finally, fort=∂/∂y_none can check, using (4.2) and (4.3), that

a^jk[y_nu_xn−x_nu_yn]_jk= 2 Im(a^nku_nk) = 0 and consider

v:=y_n(u−ϕ)_xn−x_n(u−ϕ)_yn. We will eventually obtain

(4.9) |utN(ζ)|C ζ∈∂B.

We claim that to get (3.7) it is now enough to estimate

(4.10) u_nn(ζ⁰)C.

Indeed, this combined with (4.8), (4.9) and (4.5) implies that all the eigenvalues of the real Hessian matrix D²u are bounded from above by C inB. But sinceuis in particular subharmonic, it follows that they must then be bounded from below by−(2n−1)C. It thus remains to show (4.10).

By (4.8) and (4.9) atζ₀ we may write (4.11) 1 =u_nnS_{m −1}+O(1),

where S_{m −1}(ζ⁰) =S_m−1((u_jk(ζ⁰))) and ifA is an n×n matrix then by A we denote the (n−1)×(n−1) matrix created by deleting thenth row and nth column in A. We will now use an idea from [21]. By (4.11) we may assume that the quantity S_{m −1}(ζ), ζ∈∂B, is minimized at ζ0. It is elementary to show that there exists a smooth mapping

Φ : (B∩B(ζ⁰,1))×Cⁿ→Cⁿ

such that for every z ∈ B ∩B(ζ⁰,1) the mapping Φ_z = Φ(z,·) is an orthogonal isomorphism ofCⁿ(and ofB), Φζ(ζ) =ζ⁰forζ∈∂B∩B(ζ⁰,1) and Φ_ζ⁰ is the identity. Forζ∈∂B∩B(ζ⁰,1) we then have

S_{m− 1}(ζ) =Sm−1(U(ζ)), where by (4.7)

U(ζ) =A(ζ) +u_N(ζ)I, A(ζ) =

(ϕ◦Φζ)_jk(ζ⁰)

−ϕN(ζ)I.

It is clear that||A||_C1,1(B∩B(ζ⁰,1))C. Define the (n−1)×(n−1) positive definite matrix

B₀:=D_m−1(U(ζ⁰)) =

∂Sm−1

∂a_pq (U(ζ⁰))

.

By (2.2) and (2.3) tr

B0(U(ζ)−U(ζ⁰))

S_{m −1}(ζ)−S_{m −1}(ζ⁰)0.

We thus obtain

v(ζ) :=u_xn(ζ)−u_xn(ζ⁰)+∇u(ζ), ζ−ζ⁰+(t r B₀)⁻¹tr

B₀(A(ζ)−A(ζ⁰))

0

forζ∈∂B∩B(ζ⁰,1).

Similarly as before, we define the barrierw:=v−C₁|z−ζ⁰|²+C₂ψ, and choosingC1C2 under control we getw0 on∂(B∩B(ζ⁰,1)) and a^jkw_jk0 inB∩B(ζ⁰,1). Thereforew0 inB∩B(ζ⁰,1) and

u_xnxn(ζ⁰)C

which together with (4.8) gives (4.10).

5. The class Dm.

Essentially just repeating the proof of [5, Theorem 1.1] and using the necessary machinery from Section 3, we can get the following characteriza- tion of the classDm (we consider the germs of functions).

THEOREM5.1. — For a negativeu∈ Pmthe following are equivalent i)u∈ Dm;

ii) For every sequence u_j ∈ Pm∩C^∞ decreasing tou the sequence Hm(uj)is locally weakly bounded;

iii) u∈ L^m_loc and for every sequenceuj ∈ Pm∩C^∞ decreasing to u the sequences

(5.1) |u_j|^m−p−2du_j∧d^cu_j∧(dd^cu_j)^p∧ω^m−p−1, p= 0,1, . . . , m−2, are locally weakly bounded;

iv)u∈L^m_loc and there exists a sequenceu_j∈ Pm∩C^∞decreasing to usuch that the sequences (5.1) are locally weakly bounded.

For m= 2 it is clear that conditions iii) and iv) in Theorem 5.1 are equivalent, and they mean precisely that u∈ P2∩W_loc^1,2.

We will now very briefly sketch the proof of Theorem 5.1. The crucial steps are the following two estimates.

PROPOSITION 5.2. — Let Ω Ω be domains in Cⁿ. Assume that 2 mn and that eitherr 0 or r 1. Then for any u∈ Pm∩C(Ω), u <0, we have

Ω

|u|^r(dd^cu)^m∧ω^n−mC

Ω

|u|^rdu∧d^cu∧(dd^cu)^m−2∧ω^n−m+1, where Cis a positive constant depending only on Ω andΩ.

Proof. — It is the same as the proof of [5, Proposition 2.1].

THEOREM 5.3. — Let Ω Ω be domains in Cⁿ. Assume that 2 m n and r 0. Then for u, v ∈ Pm∩C(Ω) with u v < 0 one has

Ω

|v|^rdv∧d^cv∧(dd^cv)^m−2∧ω^n−m+1

C

Ω

|u|^m+rωⁿ+

m−2 p=0

Ω

|u|^m−p+r−2du∧d^cu∧(dd^cu)^p∧ω^n−p−1

,

where Cis a constant depending only on Ω,Ωandr.

Proof. — One has to repeat the proof of [5, Theorem 2.2].

Proof of Theorem 5.1 (sketch). — It follows from Theorem 5.3 that the conditions iii) and iv) are equivalent. To show implication iii)⇒iv) one has to use Cegrell’s arguments (see the proof of [9, Theorem 4.2], they are also presented in [5]). The implication i)⇒ii) is trivial and to show the remaining implication ii)⇒iii) we proceed the same way as in [5]. We have however to show in addition thatuis inL^m_loc which is already guaranteed form=n(and it will follow from Proposition 1.3 ifm²/(m−1)> n).

Suppose that ii) is satisfied but u /∈ L^m_loc. We can then find balls B B such thatu is defined in a neighborhood of B andu /∈ L^m_loc(B).

Let vj =u∗ρ_1/j be the sequence of the regularizations of u. Then there exists an increasing sequencek=k(j)j such that

(5.2)

B

|v_j−v_k|^mdλj

We set

uj: = sup{w∈ Pm(B) :wvj in B, wvk inB}

= sup{w∈ Pm(B) :whj},

where h_j ∈ C(B) is defined by h_j = v_k in B, h_j =v_j on ∂B and h_j is harmonic in B\B. By Proposition 3.2 u_j ∈ Pm(B)∩C(B). It is clear that uj is decreasing touinB and therefore by ii) we have

sup

j

B

(dd^cu_j)^m∧ω^n−m<∞.

By Theorem 3.6 we have (dd^cu_j)^m∧ω^n−m= 0 in{u_j< v_j}, and, sinceu_j vj, from Proposition 3.4 it follows that (dd^cuj)^m∧ω^n−m(dd^cvj)^m∧ω^n−m on {uj =vj}. By another application of the assumption, this time to the sequencev_j, we obtain therefore

sup

j

B

(dd^cuj)^m∧ω^n−m<∞.

However, integrating by parts in the same way as in the proof of [3, Theorem 2.1] or [5, Proposition 3.1], using (5.2) we obtain

j

B

(vj−uj)^mdλC

B

(dd^cuj)^m∧ω^n−m

where Cis independent ofj - a contradiction.

The proof of Theorem 1.1 is now the same as the proof of [5, Theorem 1.2], whereas to show Theorem 1.2 we have to proceed as in the proof of [4, Proposition 2.2] (using Theorem 3.6).

6. The L^p-estimate.

In this section we will prove Proposition 1.3. More precisely, we will show the following estimate.

PROPOSITION6.1. — Forp < n/(n−m)and negativeu∈ Pm(B(0,2)) one has

(6.1) ||u||L^p(B(0,1/2)) C||u||L¹(B(0,2)),

where Cis a positive constant depending only on n, mandp.

Proof. — We will use similar methods as for example in [10] and [24].

Thanks to regularization we may assume thatuis smooth. By C₁, C₂, . . . we will denote constants depending only onn, mandp.

Forε >0 letGε∈ Pm∩C^∞(Cⁿ) be such thatGε=GonCⁿ\B(0, ε) andG_ε↓Gasε↓0, whereGis given by (1.1). Forw∈B(0,1/2) we have,

denoting B:=B(0,1) andu_w:=u(w− ·), C₁u(w) =

B

u_w(dd^cG)^m∧ω^n−m= lim

ε→0

B

u_w(dd^cG_ε)^m∧ω^n−m

= lim

ε→0

B

Gεdd^cuw∧(dd^cGε)^m−1∧ω^n−m +

∂B

uwd^cG∧(dd^cG)^m−1∧ω^n−m

=

B

Gdd^cu_w∧(dd^cG)^m−1∧ω^n−m +

∂B

uwd^cG∧(dd^cG)^m−1∧ω^n−m=:u1(w) +u2(w).

Sinceuis in particular subharmonic,

|u2(w)|C2

∂B

|uw|dσC3||u||L¹(B(0,2)),

it is thus enough to estimate ||u1||L^p(B(0,1/2)).

WriteG=E◦ψ. Then by Proposition 2.1 we have

(6.2)

0dd^cu_w∧(dd^cG)^m−1∧ω^n−m

= (E◦ψ)^m−2(E◦ψ ω+E◦ψ dψ∧d^cψ)∧dd^cu_w∧ωⁿ⁻² C₄|z|^{−2n(m−1)/m}∆u_w,

sinceE<0. SetG:=G_1/2, so thatG∈C^∞(B) andG=Gnear∂B, and letϕ∈C₀^∞(B(0,3/2)) be such thatϕ= 1 inB and 0ϕ1 elsewhere.

Then

B

dd^cuw∧(dd^cG)^m−1∧ω^n−m=

∂B

d^cG∧dd^cuw∧(dd^cG)^m−2∧ω^n−m

=

B

dd^cu_w∧(dd^cG)^m−1∧ω^n−m

B(0,3/2)

ϕdd^cu_w∧(dd^cG)^m−1∧ω^n−m (6.3)

=

B(0,3/2)

uwdd^cϕ∧(dd^cG)^m−1∧ω^n−m C₅||u||L¹(B(0,2)).